Euler's equations

The fundamental equation of motion of a rotating body [see Equation (8.3)],

(8.23)

is only valid in an inertial frame. However, we have seen that
is most simply expressed in a frame of reference whose axes are aligned
along the principal axes of rotation of the body. Such a frame of reference
rotates with the body, and is, therefore, non-inertial. Thus, it is helpful to define two Cartesian coordinate systems with the same origins. The first,
with coordinates
,
,
, is a fixed inertial frame; let us denote
this the fixed frame. The second, with coordinates
,
,
,
co-rotates with the body in such a manner that the
-,
-, and
-axes are always pointing along its principal axes of rotation; we shall
refer to this as the body frame. Because the body frame co-rotates with the body, its instantaneous angular velocity is the same as that of the
body. Hence, it follows from the analysis in Section 6.2 that

(8.24)

Here,
is the time derivative in the fixed frame, and
the
time derivative in the body frame.
Combining Equations (8.23) and (8.24), we obtain

(8.25)

In the body frame, let
and
.
It follows that

(8.26)

where
,
and
are the principal
moments of inertia. Hence, in the body frame, the components of Equation (8.25) yield

where
.
Here, we have made use of the fact that the moments of inertia
of a rigid body are constant in time in the co-rotating body frame.
The preceding three equations are known as Euler's equations.

Consider a body that is freely rotating; that is, in the absence of external
torques. Furthermore, let the body be rotationally symmetric
about the
-axis. It follows that
.
Likewise, we can write
. In general, however,
. Thus, Euler's equations yield

Clearly,
is a constant of the motion.
Equation (8.30) and (8.31) can be written

(8.33)

and

(8.34)

where
. As is easily
demonstrated, the solution to these equations is

(8.35)

and

(8.36)

where
is a constant. Thus, the projection of the
angular velocity vector onto the
-
plane has the fixed length
, and rotates steadily about the
-axis with angular
velocity
.
It follows that the length of the angular
velocity vector,
, is a constant of the motion.
Clearly, the angular
velocity vector subtends some constant angle,
, with the
-axis, which implies that
and
.
Hence, the components of the angular velocity vector are

(8.37)

(8.38)

and

(8.39)

where

(8.40)

We conclude that, in the body frame, the angular velocity vector precesses about the
symmetry axis (i.e., the
-axis) with the angular
frequency
. Now, the components of the angular momentum vector are

(8.41)

(8.42)

and

(8.43)

Thus, in the body frame, the angular momentum vector is also of constant length, and
precesses about the symmetry axis with the angular frequency
. Furthermore, the angular momentum vector subtends a constant angle
with the symmetry
axis, where

(8.44)

The angular momentum vector, the angular velocity vector, and
the symmetry axis all lie in the same plane;
that is,
,
as can easily be verified. Moreover, the
angular momentum vector lies between the angular velocity vector and
the symmetry axis (i.e.,
) for a flattened (or oblate) body
(i.e.,
), whereas the angular velocity vector lies
between the angular momentum vector and the symmetry axis (i.e.,
) for an elongated (or prolate) body (i.e.,
). (See Figure 8.2.)